A B S T R A C T S - pdfs.semanticscholar.org · Zbaganu, G. Branching Processes and Insurance...

XII-th International Summer Conference

on Probability and Statistics (ISCPS)

Seminar on Statistical Data Analysis

(SDA’2006)

A B S T R A C T S

Sozopol, Bulgaria 17-25 June, 2006

Contents

Alawneh, A. Calculation of Steady- State Probabilities of M/M Queues . . . . . . . . .5Alsaidi, A. Statistical Analysis of the Influence of Explicit vs. Implicit InstructionalApproaches during a Technology-based Curriculum on Students’ Understanding ofNature of Science (NOS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Al-Saidy, O., AL-Samarraie, A. Regression Estimators Using Extreme RankedSet Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Atanasov, D. A Robust Modification of Metric Unfolding Procedure . . . . . . . . . . . . 6Atanasov, D., Stoimenova, V., Yanev, N. Estimators in Branching Processeswith Immigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Belaide, K., Bentarzi, M. The Principles Properties of Periodic Fractionary Stochas-tic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Benchettah, A. A Stochastic Control Approach to a Fokker-Planck Equation, Re-ciprocal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Boscaiu, V. Residuals Analysis of some GLM models . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Bulinskaya, E. Sensitivity Analysis of Some Applied Probability Models . . . . . . . . 9Capron, X., Massart, D.L., Smeyers-Verbeke, J. Authentication of the Originof Wines by Multivariate Discrimination Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Christozov, D., Mateev, P., Mutafchiev, L. Statistical Expertise Needed toTrain Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Chukova, S., Hayakawa, Yu, Johnston, M. Warranty Analysis: Warranty Re-pair Strategy based on the degree of repair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Dabala, W., Lednicki, B. Estimation of Fraction in Population based on PartiallyRealized Random Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Dias, G., Santos, C., Cleuziou, G. Automatic Knowledge Representation usinga Graph-based Algorithm for Language-Independent Lexical Chaining . . . . . . . . . . . 13Donchev, D. An Excursion Characterization of the First Hitting Time of BrownianMotion in a Smooth Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Donchev, D., Kralchev, D. On the Moving Boundary Hitting Probability for aBrownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Ford, K. Probability Estimates for Random Walks with Barriers . . . . . . . . . . . . . . . 14Gautherat, E., Bertail, P., Harari-Kermadec, H. Empirical Discrepancies andQuasi-Empirical Likelihood : Exponential Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Gayraud, G., Gautherat E. Parametric Estimation in Noisy Blind Deconvolution:a New Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Georgieva, R., Atanassov, E., Gurov, T., Ivanovska, S., Karaivanova, A.,Nedjalkov, M. SALUTE – GRID Application for problems in quantum transport.Part I: Importance separation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16Gonzalez, M., Martinez, R., Slavtchova-Bojkova, M. Stochastic Monotonyand Continuity Properties for the Extinction Time of Age-Dependent Branching Pro-cesses: an Application to Epidemic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Gueorguieva, R. Correlated Probit Models for Joint Analysis of Repeatedly Mea-sured Binary, Ordinal and Continuous Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Haynatzki, G., Haynatzka, V., Brand, R., Sherman, S., Lynch, H., Recker,R., Statistics for Genetic Anticipation with Applications . . . . . . . . . . . . . . . . . . . . . . . 17Ivanovska, S., Atanassov, E., Georgieva, R., Gurov, T., Karaivanova, A.,

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Nedjalkov, M. SALUTE – GRID Application for problems in quantum transport.Part II: Hybrid algorithms and parallelization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Jacob, C., Viet, A.-F. A New Class of Stochastic Processes for Formalizing andGeneralizing Population-dependent Individual-based Branching Models: the Semi-Semi-Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Jaradat, A., Smadi, M. Asymptotic Efficiencies of the Survival Functions Estima-tors for the Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Jordanova, P. (G, λ)-extremal Processes and their Connection with Max-stable Pro-cesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Kalinov Testing for (Bio)Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19Kazemnejad, A., Akhoond, M.R. Application of Mixture Regression Model inModeling Length of Stay in Hospital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Kharin, A. Robustness in Sequential Testing of Hypotheses on Parameters of Ran-dom Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20Kharin, Y. Discrete Time Series with “Long Memory” . . . . . . . . . . . . . . . . . . . . . . . . . 21Klebanov, L., Yakovlev, A. A New Type of Stochastic Dependence Revealed inGene Expression Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Kolkovska, E. Occupation Measure of Classical Risk Processes . . . . . . . . . . . . . . . . .23Maillard-Teyssier, L., Jacob, C., Denis, J.-B. A semi-Markovian BranchingProcess on the Clinical Cases for Modelling the Evolution of a SEIR Disease in aLarge Population. Example of the BSE Epidemics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Malugin, V. Estimation and Testing a Nonlinear Transformation of the IntegratedProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Markovski, S., Bakeva, V. Uniformity obtained by Quasigroups: Part II . . . . . 25Mihova, M., Popeska, Z. Multistate Systems with Graduate Failures and EqualTransition Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Milanov, V., Nickolov, R. Entropy Based Approach to Finding Interacting GenesResponsible for Complex Human Diseases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Milenkov, K., Daskalova, N. A Simple Kernel-based Parameter Estimation . . 26Mimbela, J.A.L. Finite Time Blowup and Stability of a Reaction - Diffusion Equa-tion with a Time-dependent Levy Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Minkova, L. Reinsurance by the Polya - Aeppli Risk Model . . . . . . . . . . . . . . . . . . . . 28Mitov, G., Mitov, K. An Option Pricing Formula based on Branching Processes28Mitov, I., Pancheva, E. Sum and Extremal Processes over Explosion Area . . . .28Mota, M., Gonzalez, M., Martinez, R. Controlled Multitype Branching Models:Geometric Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Mouhoubi, Z., Aissani, D. Stability of the Inventory - Backorder Process in the(R,S) Inventory/Production Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Moussas, C., Noncheva, V. Extraction of Fraud Schemes from Trade Series . . 30Mutafchiev, L., Kamenov, E. The Number of Parts of Given Multiplicity in aRandom Integer Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Neykov, N., Filzmoser, P., Dimova, R., Neytchev, P. Robust Fitting of Mix-ture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31Neykov, N., Zucchini, W., Neytchev, P., Hristov, H. Modelling Daily Precip-itation over the Territory of Bulgaria using Hidden Markov Models . . . . . . . . . . . . . . 32Nickolov, R., Milanov, V. A Test of Association Between Qualitative Trait and a

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Set of SNPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32Oja, H., Sirkia, S., Eriksson, J. Scatter Matrices, Kurtosis and Independent Com-ponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Ozdemir, A.I. A Comparative Performance Analysis of the Turkish Industrial Sec-tors in Terms of ERP Success . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Popovic, B., Stojanovic, V. Generalized Split-BREAK Process . . . . . . . . . . . . . . 34Romisch U., Jager, H. Vandev, D. Interactive Regularized Discriminant Analysisfor Discriminating Authentic Wines from five Countries . . . . . . . . . . . . . . . . . . . . . . . . . 34Ramos, A., Molina, M., del Puerto, I. Some Probabilistic Results in a BisexualBranching Process with Immigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Sagitov, S. Size-biased Branching Processes with Overlapping Generations . . . . . 36Shishkov, B. On the Use of Higher-Order Statistical Tests in Signal Processing 36Stoev, S., Taqqu, M. Limit Theorems for Maxima of Heavy-tailed Terms withRandom Dependent Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Stoimenova, E. Smoothed Density Estimation of Interval Censored Data . . . . . . 37Stoynov, P. Levy Processes and some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . .37Stoynov, Y., Stoimenova, E. Histogram Density Estimator for Censored Data 38Toncheva, N. Two Random Variables Dependence Function . . . . . . . . . . . . . . . . . . . 38Tsitovich, I., Naumova, E., Tsitovich, F. Markov Chain Order Estimating andits Applications in Bioinformatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Tsitovich, I., Bubnov, Y., Melik-Gaykazova, E. On Robust Models of Multy-service System Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Veleva, A. An Educational Technology for the Formation of Transcendental Think-ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39Veleva, E. Joint Densities of Correlation Coefficients for Samples from MultivariateStandard Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Wakeel, M. Linear Regression with Poisson Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40Yakovlev, A., Yanev, N. Branching Processes and Cell Proliferation with Contin-uous Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Yanev, G. On Extreme Value Results in Branching Processes . . . . . . . . . . . . . . . . . . 40Yarovaya, E. Branching Symmetric Random Walk on d-dimensional Lattice . . . 41Zbaganu, G. Branching Processes and Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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Calculation of Steady-State Probabilities of M/MQueues

Ameen Alawneh

Department of Mathematics and StatisticsJordan University of Science & Technology

Irbid- Jordan 22110

e-mail: [email protected]

Consider M/M queues where the arrival rate is Poisson distributed, i.e. the dis-tribution of the time between successive arrivals is exponential, and the service timedistribution is also exponential. Calculating the steady state probabilities and thenfinding the performance measures is not an easy task. Many techniques are used inthe literature to calculate the steady state probabilities of such systems. Marks(1973)used the direct approach to solve the finite difference equations of M/M/1 queueswith priorities. Heyman (1991) presented a numerical technique based on consideringthe queue as a Markov chain then use an appropriate technique to compute explic-itly the steady state probabilities. Alawneh (1997) generalized the technique givenby Heyman (1991) to the case of two-dimensional Markov chain and successfully ap-plied the new technique into two different queuing systems. Two parallel queues andtwo priority levels without preemptive. Pasternack and Drezener (1998) presenteda recursive approach which avoids some computational difficulties in calculating thesteady state probabilities. Recently, Smith (2002) introduces a new approach. In thispaper we will make a comparison between all these techniques.

Keywords: Steady State, Queuing Systems

Statistical Analysis of the Influence of Explicit vs.Implicit Instructional Approaches during aTechnology-based Curriculum on Students’Understanding of Nature of Science (NOS)

Ahmed Alsaidi

We examine the effect of an explicit versus an implicit instructional approach dur-ing technology-based curriculum on students’ understanding of the nature of science(NOS) using advanced statistical methods. The imperative finding of the presentstudy provides evidence that teaching the NOS could be achieved through short-intensive discussion and does not necessarily require separate and independent courses.Details of the findings will be given in the talk.

Keywords: Explicit instructional approach, Implicit instructional approach, NOH

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Regression Estimators Using Extreme Ranked SetSampling

Obaid M. Al-Saidy Hani Samawi and Ahmed AL-Samarraie

Sultan Qaboos University,College of Science, Dept. of Math. and Stat. P.o.box 36, Pc 123,

Al-Khod, Oman

Regression is used to estimate the population mean of the response variable, Y ,in the two cases where the population mean of the concomitant (auxiliary) variable,X, is known and where it is unknown. In the latter case, a double sampling methodis used to estimate the population mean of the concomitant variable. We discussthe performance of the two methods using extreme ranked set sampling (ERSS).Theoretical and Monte Carlo evaluation results as well as an illustration using actualdata are presented. The results show that if the underlying joint distribution of Xand Y is symmetric, then using ERSS to obtain regression estimates is more efficientthan using ranked set sampling (RSS) or a simple random sampling (SRS).

Keywords: Efficiency, Extreme ranked set sampling, Ranked set sampling, Regres-sion Estimators

A Robust Modification of Metric UnfoldingProcedure

Dimitar Atanasov

Faculty of Mathematics and Informatics, Sofia University

The unfolding analysis is wildly used in the field of psychology for studying thepreference choice data. The metric unfolding is very sensitive to the presence of out-liers or errors in the data. Common solution for overcoming this is using of nonmetrictechniques for unfolding. The main problem arising with this is definition of order inmultidimensional space. In this paper a method, overcoming influence of outliers anderrors, for multidimensional metric unfolding is proposed. Its robust properties werestudied using the theory of d-fullness.

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Estimators in Branching Processes withImmigration

Dimitar Atanasov1 Vessela Stoimenova1 Nikolay Yanev2

1Faculty of Mathematics and Informatics, Sofia University2Department of Probability and Statistics, Institute of Mathematics and Informatics,

Bulgarian Academy of Sciences, 8, G. Bonchev, Sofia 1113, Bulgaria

In the present paper we consider the branching process with immigration andits relationship to the Bienayme - Galton - Watson process with a random numberof ancestors. Several estimators of the immigration component are considered - theconditional least squares estimator of Heyde - Seneta, the conditional weighted leastsquares estimator of Wei - Winnicki and the estimator of Dion and Yanev. Theircomparison is based on simulations of the entire immigration family trees and com-putational results. The asymptotic normality of the estimator of Dion and Yanev iscombined with the general idea of the trimmed and weighted maximum likelihood. Asa result, robust modifications of the immigration component estimator is proposed.They are based on one and several realizations of the entire family tree and are studiedvia simulations and numerical results.

2000 Math. Subj. Classification code: 60J80The paper is supported by the National Science Fund of Bulgaria, Grant No VU-MI-105/2005.

The Principles Properties of Periodic FractionaryStochastic Models

Belaide Karima and Bentarzi Mohamed

Department of Mathematics

The fractionally integrated autoregressive moving average (ARFIMA) models hasrecently received considerable attention in economic, but also in other research. AFRIMAprocesses generalize linear ARIMA models by allowing for no integer differencing pow-ers. Non stationary periodically correlated processes find their interesting applicationsin many fields such as economic, as well as hydrology and environmental studies. Itis observed that many times series which come upon in practice exhibit periodicalautocorrelated structures. Consequently, much attention has been given recently toperiodic autoregressive moving average processes. Most of the existing periodic liter-ature is concerned with identification, estimation and testing problems. On the otherhand, the invertibility propriety of periodic moving average models which its primaryimportance has been studied by Cipra (1985) an Ghysels and hall (1992). The presentwork concerns the study of periodic fractionally processes, with one order. In firstsection, we consider periodic FAR(1), given by stochastic equation (1 − at)dXt =t

(label)1(/label) where the parameter at is periodic with period S. A sufficient con-dition of causality and invertibility has been given Through section 2, .we establish

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the principles proprieties which characterized the process (autocovariance function,generatrice function of the autocovariance and autocorrelation function), we also givethe linear recurrence formula satisfying by autocovariance function. We then showthat the considered process is correlated periodically.

Keywords: Autocovariance, Autocorrelation, Fractionnary Stochastic Models

A Stochastic Control Approach to a Fokker-PlanckEquation, Reciprocal Processes

Azzedine Benchettah

Universite Badji Mokhtar, BP 12, 23000, Annaba, Algerie

We discuss relations between a controllability problem for a Fokker-Planck equa-tion, a reciprocal processes and a problem of minimum entropy: A controllabilityproblem for a Fokker-Planck equation is considered. A solution (v∗,Φ∗) to that prob-lem is constructed by a variant of Jamison’s theorem, under proper assumptions. Wegive a sufficiency condition concerning the initial and terminal data for that solutionto exist. We show that v∗ is an optimal feedback control for a stochastic optimalcontrol problem and corresponds also to a solution of a problem of minimum entropy.Further, we prove that the corresponding optimally controlled stochastic process is areciprocal process which is Markov.

Keywords: Fokker-Planck equation, reciprocal processes, minimum entropy dis-tance, stochastic optimal control2000 Mathematics Subject Classification: 49L20, 60J60, 93E20

Residuals Analysis of some GLM models

Voicu Boscaiu

Institute of Mathematical Statistics and Applied Mathematics, Bucharest

Why the analysis of residuals of some different general linear models offers similar(practically speaking) conclusions? Why, often, the conclusions can be “reasonableaccurate”, doesn’t matter the hypotheses of the considered models can’t be simulta-neously true? We try to give some answers to the above questions.

Keywords: residuals, linear regression, empirical analysis, models comparison

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Sensitivity Analysis of Some Applied ProbabilityModels

Ekaterina Bulinskaya

Moscow State University

The aim of any applied probability researcher is investigation of systems perfor-mance and its optimization. To this end one can use a lot of mathematical modelsmore or less precisely describing the system. How to choose an appropriate model?Firstly, it is necessary to decide what is more important at the moment, namely,precision of a model or its solvability (see e.g. (1)). Thus, one has always to reckonwith errors in the results of investigation and, hence, in systems management. Theerrors may be due to poor fitness of the model. On the other hand, the model usuallydepends on parameters which are estimated on the base of previous observations ofsystems functioning. That is the second source of errors. So it is important to mea-sure the influence of input parameters deviations on the output fluctuations. Thatis the task of the global sensitivity analysis. The main attention in presentation willbe focused on Sobol’ method of function and variance decomposition (see e.g., (5)),as well as FAST method (see e.g., (4)), and their application to some models arisingin insurance and inventory theory (see e.g. (2)). The third type of errors, associatedwith randomness of underlying processes, will be dealt with using risk measurement(see e.g. (3)).The research was partially supported by the RFBR grant 05-01-00256.References:(1) Bellman R. Dynamic Programming. Princeton University Press. Princeton. 1957.(2) Bulinskaya E. On decision making under uncertainty. In: Proceedings of the 5thSt.Petersburg Workshop on Simulation. June 26 - July 2, 2005, 181-186.(3) McNeil A.J., Frey R., Embrechts P. Quantitative Risk Management. PrincetonUniversity Press. Princeton. 2005.(4) Saltelli A., Tarantola S., and Chan K.P.-S. A Quantitative model independentmethod for global sensitivity analysis of model output. Technometrics, 1999, 41, 1,39-56.(5) Sobol’ I.M. Sensitivity estimates for nonlinear mathematical models. Matem.Modelirovanie, 1990, 2(1), 112-118. (in Russian)

Keywords: Input-Output models, Global Sensitivity Analysis, Risk Measurement,Insurance

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Authentication of the Origin of Wines byMultivariate Discrimination Methods

X. Capron D.L. Massart and J. Smeyers-Verbeke

Pharmaceutical Institute, Vrije Universiteit Brussel, Laarbeeklan 103, 1090 Brussels,

Belgium.

In the context of the European WineDB project ”Establishing of a wine databank for analytical parameters for wines from Third countries (G6RD-CT-2001-00646-WINE DB)” funded by the European Commission within the fifth framework ”Com-petitive and Sustainable Growth”, a data base containing 1200 samples of authenticand commercial wines from Hungary, Czech Republic, Romania and South Africa hasbeen created. For each of those samples the data base contains around 100 parameterssuch as the concentration for some trace and macro elements, classical parameters orisotopic ratios (63 variables), as well as information about the geographical origin ofthe sample. The goal of the project is to evaluate whether it is possible to determinethe country of origin of a wine using its chemical content. Multivariate analysis toolssuch as Partial Least Squares (PLS) regression or Classification and Regression Trees(CART) are considered here. The discriminating power of the different parametersavailable is also evaluated. To do so, variable selection methods such as PLS-UVE (foruninformative variable elimination) are applied in order to retain only the parametersimportant for the discrimination. The results obtained showed that it is possible todiscriminate wine samples very efficiently with the PLS approach. Indeed, around 10variables are required to determine correctly the origin of authentic wines 95% of thetime, though the set of necessary variables is depending on the vintage and country oforigin of the considered sample. A kernel Support Vector Machine approach (SVM)is then used to improve those results. Kernel SVM models using a unique set of sixparameters could authenticate the authentic wine samples with a minimum successrate of 94%. The discrimination of commercial samples is not as straightforward andthe models built require at least 25 variables. Those models could determine thecountry of origin of a wine without error more than 90% of the time.

Statistical Expertise Needed to Train Data Mining

D. Christozov1 P. Mateev2 and L. Mutafchiev1

1American University in Bulgaria2FMI, Sofia University ”St. Kl. Ohridski”

Statistical expertise is essential in building knowledge and skills needed to developand use modern information applications, developed to serve researchers in differentareas such as natural sciences, business, economics, etc. Whether the curriculumof those disciplines includes training the needed body of statistical knowledge anddeveloped the needed skills to perform statistical inquires? The paper addresses anarrow area of needs of statistical knowledge – the area of training Data Mining,which is recommended by ACM MSIS. The research includes matching the list of

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statistically dependent topics from a standard Data Mining course with knowledgeprovided in the required statistical courses in both Sofia University and AmericanUniversity in Bulgaria. This research may serve to better address the topics in theData Mining course, which are not well covered in a general statistical course and toadjust the content of the general statistical courses to meet the further needs.

Keywords: Statistical Education, Data Mining

Warranty Analysis: Warranty Repair Strategybased on the degree of repair

S. Chukova1 Yu Hayakawa2 and M. Johnston1

1School of Mathematics, Statistics and Computer Science,Victoria University of Wellington, PO Box 600, Wellington, New Zealand

e-mail: stefanka, [email protected] of International Liberal Studies, Waseda University,1–21–1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-0051, Japan


An approach to modeling imperfect repairs under warranty settings using the con-cepts of delayed and accelerated distribution functions is presented. A procedure ofestimating the degree of repair as well as other modeling parameters by Markov chainMonte Carlo methods is designed. We focus on a particular warranty repair strategy,related to the degree of the warranty repair, for a nonrenewing, two-dimensional, freeof charge to the consumer warranty policy. We consider a rectangular warranty regionand divide it into disjoint subregions, so that, for a faulty item, each of these subre-gions has a preassigned degree of repair. Our main goal is to determine the subregions,so that the associated expected warranty servicing cost per item sold is minimised. Acomparison between our strategy and previously studied, more restrictive, strategiesis provided.

Estimation of Fraction in Population based onPartially Realized Random Sample

Wieslawa Dabala1 and Bronislaw Lednicki2

1Public Opinion Research Centre, Warsaw2Central Statistical Office, Warsaw

Sampling was a three-stage process with stratification. The Hartley-Rao proce-dure was used to select the first-stage units. Second-stage units were selected from theunits of the first stage in a procedure without replacement. Next, the Kisch methodwas used to select respondents from the units selected at the second stage.The realized sample size was smaller than the selected sample size.Let:U – finite population, U = u1, u2, . . . , uN,

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N – number of units in population U ,Ud – non-empty subset of U separated for analysis, or in special case, the whole pop-ulation,Nd – number of units in Ud,d = 1, 2, . . . , D,D – number of subsets in analysis,X, Y – functions defined on U with values 1 or 0.The essential estimation parameter of the population is the fraction of persons witha distinguished property:

Rd =

Nd∑i=1

Xi

Nd∑i=1

Yi

, where:

Xi =

1 for ui ∈ Ud with distinguished property0 for other units

Yi =

1 for ui ∈ Ud

0 for other units

Lets∗d – the total sample sampled from Ud (s∗d ⊂ Ud),sd – total realized sample from Ud (sd ⊂ s∗d ⊂ Ud),nd – the realized sample size from Ud.

The estimator of parameter Rd is rd: rd =

nd∑i=1

wixi

nd∑i=1

wiyi

, where:

xi =

1 for ui ∈ sd with distinguished property0 for other units

yi =

1 for ui ∈ sd with distinguished property0 for other units

wi – weight for ui ∈ sd.The calculation of weights included:– the probability of selecting of a unit at each stage,– response rate and the imperfections of the sampling frame,– Central Statistical Office data on the population structure.The complex sampling design and the need to use weights result in complex r esti-mator, and thus in the need for simplified methods to evaluate its variance as in thiscase Jackknife.

Keywords: applications of mathematical statistics, applications of representativemethod in public opinion po

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Automatic Knowledge Representation using aGraph-based Algorithm for Language-Independent

Lexical Chaining

Gael Dias Claudia Santos and Guillaume Cleuziou

University of Beira Interior

Lexical Chains are powerful representations of documents. In particular, theyhave successfully been used in the field of Automatic Text Summarization. How-ever, until now, Lexical Chaining algorithms have only been proposed for English.In this paper, we propose a greedy Language-Independent algorithm that automati-cally extracts Lexical Chains from texts. In particular, we propose to automaticallyconstruct from a collection of documents a lexico-semantic knowledge base with thepurpose to identify cohesive lexical relationships between words based on corpus evi-dence. This hierarchical lexico-semantic knowledge base is built by using the PoBOC(Pole-Based Overlapping Clustering) Algorithm (Cleuziou et al., 2004) that clusterswords with similar meanings and allows words with multiple meanings to belong todifferent clusters. The second step of the process aims at automatically extractingLexical Chains from texts based on our knowledge base. For that purpose, we proposea new greedy algorithm which can be seen as a mixture of (Barzilay and Elhadad,1997) and (Hirst and St-Onge, 1997) algorithms and which implements (Lin, 1998)’sinformation-theoretic definition of similarity as the relatedness criterion for the attri-bution of words to Lexical Chains. As a consequence, our methodology can be appliedto any language and proposes a solution to language-dependent Lexical Chainers

Keywords: Knowledge Representation, Natural Language Processing, Graph-basedOverlapping Clustering

An Excursion Characterization of the First HittingTime of Brownian Motion in a Smooth Boundary

Doncho S. Donchev

FMI, Sofia University

We obtain a formula for the distribution of the first hitting time of Brownianmotion in an one-sided curved boundary f(t) which gives a new characterization ofthe first-passage density p(t). It is shown that p(t) is a product of the known densityfor reaching the level y = f(0) at time t and a function that is a solution to theCauchy problem for some parabolic operator. The boundary-independent part ofthis operator is just the Kolmogorov’s equation for a Brownian excursion transitiondensity.

Keywords: Boundary crossing, Brownian excursion, Girsanov’s transform, condi-tional processes

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On the Moving Boundary Hitting Probability for aBrownian Motion

Doncho S. Donchev1 and Dobromir P. Kralchev2

1Sofia University2University of Food Technologies - Plovdiv

Consider the probability that a Brownian motion hits a moving boundary by acertain moment conditioned that its sample path stays above a fixed level. In case ofa linear boundary we find a formula for this probability.

Keywords: Brownian motion, hitting time, Laplace transform

Probability Estimates for Random Walks withBarriers

Kevin Ford

University of Illinois at Urbana-Champaign Department of Mathematics 1409 West Green

St. Urbana, IL 61802, USA

We consider a recurrent random walk S0, S1, . . . where Sj = X1 + ...+Xj and theXj are i.i.d. random variables with mean 0 and variance 1. We prove very preciseestimates for the probability that Sj ≤ y(j = 0, 1, ..., n− 1) given that Sn = x, undergeneral conditions on the distribution of Xi. We will discuss an application of thesebounds to the theory of order statistics, giving a sharpening and generalization of anestimate of Smirnov from 1939.

Keywords: random walk, order statistics

Empirical Discrepancies and Quasi-EmpiricalLikelihood : Exponential Bounds

Emanuelle Gautherat1 Patrice Bertail2 and Hugo Harari-Kermadec2

1CREST-LS, University of Reims, France2CREST-LS, INRA, University of Nanterre

We study some extensions of the empirical likelihood method, when the Kullbackdistance is replaced by some general convex divergence or discrepancy. We proposeto use, instead of empirical likelihood, some regularized form or quasi-empirical like-lihood method, corresponding to a convex combination of Kullback and χ2 discrep-ancies. We show that for some adequate choice of the weight in this combination, thecorresponding quasi-empirical likelihood is Bartlett correctable. We also establishsome non-asymptotic, explicit and exponential bounds for the confidence intervalsthat may be deduced by using this method. These bounds are derived via the study

14

of self-normalized sums in the multivariate case. The results on self-normalized sumsare of interest by themselves.Jing, Bing-Yi and Wang, Qiying, 1999, An Exponential nonuniform Berry-Esseenbound for self-normalized sums, Annals of Probability, vol = 27, n 4, 2068-2088.Jing, Bing-Yi and Shao, Qi-Man and Wang, Qiying, 2003, Self-Normalized Cramer-Type Large Deviations for independent random variables, Annals of Probability, vol=31, n 4,2167-2215.Panchenko, Dmitry, 2003, Symmetrization approach to concentration inequalities forempirical processes, Annals of Probability, vol = 31, n 4, 2068-2081.Chistyakov, G. P. and Gotze, F., 2003, Moderate Deviations for Student’s Statistic,Theory of Probability & Its Applications, vol = 47, n 3, 415-428.Owen, A. B., 1990, Empirical likelihood ratio confidence regions, Annals of Statistics,vol = 18, 90-120.

Keywords: Empirical likelihood, self-normalized process, exponential bounds, Ho-effding inequality

Parametric Estimation in Noisy BlindDeconvolution: a New Estimation Procedure

Ghislaine Gayraud1 and E. Gautherat2

1University of Rouen, and LS-CREST2Economic Faculty of Reims, and LS-CREST

In the framework of noisy blind deconvolution parametric systems in the complexcase, our main contribution is to provide a new estimation procedure for the inversefilter. This method differs from other related papers since our procedure estimationis based on the zeros of a criteria function whereas it is classically based on minimiz-ing a penalized empirical contrast function. We also give estimators for the level ofnoise and the law of the input signal. A consistency result for all estimates and alimit distribution theorem for the level noise and the inverse filter are established. Aconsistent simulation study is added in order to demonstrate empirically the compu-tational performance of our estimation procedures.

REFERENCES:R. Chen and T. H. Li (1995), Blind Restoration of Linearly Degraded Discrete Signalsby Gibbs Sampler, IEEE Transactions on Signal Processing, 43, 2410-2413.F. Gamboa and E. Gassiat (1996), Blind Deconvolution of discrete linear systems,Annals of Stat., 24, 1964-1981.E. Gassiat and E. Gautherat (1998), Identification of noisy linear systems with discreterandom input, IEEE Transaction on Information Theory, 44, 1941-1952.E. Gassiat and E. Gautherat (1999), Speed of convergence for the blind deconvolutionof a linear systems with discrete random input, Annals of Stat., 27, 1684-1705.E. Gautherat (2002), Deconvolution aveugle bruitee: estimation de la distribution du

15

processus source, Preprint-INSEE-Lucarnes Bleues , 2002-22.T. H. Li (1995), Blind deconvolution of linear systems with multilevel nonstationaryinputs, Annals of Statistics, 23, 690-704.

Keywords: Noisy deconvolution system, blind identification, inverse filter, Tchebychev-system, spatial model.

SALUTE – GRID Application for problems inquantum transport. Part I: Importance separation

algorithms

R. Georgieva E. Atanassov T. Gurov S. Ivanovska A. Karaivanova and M.Nedjalkov

Institute for Parallel Processing - Bulgarian Academy of Sciences, Sofia, Bulgaria

SALUTE (Stochastic ALgorithms for Ultra-fast Transport in sEmiconductors)is a Grid application developed for solving computationally intensive problems inquantum transport. Formally, it consists of a bunch of Monte Carlo algorithms forsolving integrals and integral equations. Monte Carlo (MC) methods for quantumtransport in semiconductors and semiconductor devices have been actively developedduring the last decade. If temporal or spatial scales become short, the evolution ofthe semiconductor carriers cannot be described in terms of the Boltzmann transportand therefore a quantum description is needed. As a rule quantum problems arevery computationally intensive and require parallel and Grid implementations. Inthis paper we present description of the problem, physical model and Monte Carloapproach. In details the importance separation is described which is included inSALUTE in order to improve the convergence rate.

Stochastic Monotony and Continuity Properties forthe Extinction Time of Age-Dependent BranchingProcesses: an Application to Epidemic Modelling

Miguel Gonzalez1 Rodrigo Martinez1 and Maroussia Slavtchova-Bojkova2

1Department of Mathematics. University of Extremadura. 06011-Badajoz, Spain

e-mail: [email protected], [email protected] of Probability and Statistics. Institute of Mathematics and Informatics,

Bulgarian Academy of Sciences, 1113-Sofia, Bulgaria


Strongly motivated by the need of models to accurately study the natural and”man-made” spread of infectious diseases in human and/or animal populations wedraw our efforts to the well-known age-dependent branching process as an appropriatetool in doing such analysis. For this model we study stochastic monotony and conti-nuity properties for its extinction time depending on the reproduction law. Finally we

16

model the spread of an infectious disease into a population by way of age-dependentbranching processes and apply the obtained theoretical results to determine optimalvaccination policies in order to eradicate the disease.The research was partially supported by the Ministerio de Ciencia y Tecnologia andthe FEDER through the Plan Nacional de Investigacion Cientifica, Desarrollo e In-novacion Tecnologica, grant BFM2003-06074 and by NFSI, grant VU-MI-105/2005,Bulgaria.

Keywords: Age-dependent branching process. Epidemic models

Correlated Probit Models for Joint Analysis ofRepeatedly Measured Binary, Ordinal and

Continuous Outcomes

Ralitza Gueorguieva

Yale University

In longitudinal studies and in clustered situations often multiple discrete and con-tinuous outcome variables are measured for each individual. Joint analysis of theresponses results in better control of type I error, allows for estimation of overalltreatment effect, may provide additional information about the correlation structureof the data and may lead to greater statistical power. Models for such situations arenecessarily complex since both correlations between repeated measures on the samevariable, and correlations between different variables must be taken into account. Wepropose correlated probit models for binary, ordinal and continuous cluster-level andindividual-level variables. We demonstrate how to perform maximum-likelihood esti-mation and how to use likelihood ratio tests for model selection. Bias and efficiencyunder model misspecification are assessed via simulations.

Statistics for Genetic Anticipation withApplications

Gleb Haynatzki, PhD Vera Haynatzka, PhD, Randolf Brand, MD, Simon Sherman,PhD, Henry Lynch, MD, Robert Recker, MD

Creighton U, U of Chicago, UNMC

Genetic anticipation for a particular disease can involve an earlier age at on-set/occurrence (which is our focus here), greater severity, and/or a higher numberof affected individuals in successive generations within a family. Nonparametric teststatistics are investigated for matched and unmatched data. Comparison of ages ofdiagnosis/occurrence among different birth cohorts, before and after adjustment fortime under observation, is the main focus of this presentation. Applications to familialpancreatic cancer and fragility bone fracture are used as illustrations of the proposedstatistical methodology.

Keywords: anticipation, genetic epidemiology, pancreatic cancer, bone fracture

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SALUTE – GRID Application for problems inquantum transport. Part II: Hybrid algorithms and

parallelization.

S. Ivanovska E. Atanassov R. Georgieva T. Gurov A. Karaivanova and M.Nedjalkov

Department of Parallel Algorithms, Institute for Parallel Processing - Bulgarian Academy

of Sciences

SALUTE is a pilot grid application where the stochastic approach relies on thenumerical MC theory applied to the integral form of the generalized electron-phononWigner equation. In this paper we present a new version of SALUTE where hybridalgorithms are added in order to reduce the computing time. Monte Carlo applica-tions are widely perceived as computationally intensive but naturally parallel. Thesubsequent growth of computer power, especially that of the parallel computers anddistributed systems, made possible the development of distributed MC applicationsperforming more and more ambitious calculations. Compared to the parallel com-puting environment, a large-scale distributed computing environment or a Compu-tational Grid has tremendous amount of computational power. Let us mention theEGEE Grid which today consists of over 18900 CPU in 200 Grid sites. By usingthe Grid environment provided by the EGEE project middleware, we were able toreduce the computing time of Monte Carlo simulations of ultra-fast carrier transportin semiconductors.

A New Class of Stochastic Processes forFormalizing and Generalizing Population-dependent

Individual-based Branching Models: theSemi-Semi-Markov Processes

Christine Jacob and Anne-France Viet

National Agronomical Research Institute

Individual-based models are a “bottom-up” approach for calculating empirical dis-tributions at the scale of the population from simulated individual trajectories. Webuild a new class of stochastic processes for mathematically formalizing and generaliz-ing these simulation models according to a “top-down” approach. We allow individualpopulation-dependent semi-Markovian transitions in an evoluating population suchas a branching population. These new processes are called Semi-Semi-Markov Pro-cesses (SSMP) and are generalizations of Semi-Markov processes. We calculate theirkernels, their marginal probability law and their backward equations. Examples aregiven.

Keywords: Individual-based model; Branching process; semi-Markov; semi-semi-Markov

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Asymptotic Efficiencies of the Survival FunctionsEstimators for the Exponential Distribution

Ahmed A. Jaradat1 and Mahmoud M. Smadi

1Jordan University of Science and Technology

In medical statistics, the survival function is a relationship between proportionand time. Proportion is the proportion of subjects which are still surviving at time,t. The term is also used in other fields and is known as ”units still operating” insteadof subjects still alive. In this paper, an estimator of the survival time, Xi, for theith patient on a clinical trial with censoring time, Ti (dropping out of the trial) andits properties, when both survival and censoring time are exponentially distributed,considered. A simulation is carried out to determine the performance of the estimatorsfor different combinations of parameters related to the survival and censoring times.

Keywords: Survival function, time censoring, root mean square errors (RMSE)simulation.

(G, λ)-extremal Processes and their Connection withMax-stable Processes

Pavlina Jordanova

Faculty of Mathematics and Informatics, Shumen University, Bulgaria

The study of G-extremal processes was initiated by S.Resnick and M. Rubinovich(1973). Here we transform these processes by strictly increasing and continuous func-tion λ : (0,∞) → (0,∞) and investigate the connection between (G, λ)- extremalprocesses and max-stable extremal processes. We prove that about the processeswith independent max-increments if one of the following three statements is given,the other two are equivalent: a) G is a max-stable process; b) Y is (G, λ)- extremalprocesses; c) Y is self-similar extremal process.

Keywords: G-extremal processes, max-stable processes, self-similar processes

Testing for (Bio)Equivalence

Krassimir Kalinov

Department of Computer Science, New Bulgarian University, 21 Montevideo Str., 1618

Sofia, BULGARIA

Failure to reject the null hypothesis of equality does not imply equivalence betweentwo population means. The fact that the lack of evidence is not the evidence of lackis especially important for pharmaceutical industry when it is necessary to proof the

19

similar behavior of two medicines. The common pharmacokinetic measures in statis-tical proving of (bio)equivalence are the area under curve (AUC) and the maximalconcentration (Cmax). Usually these two parameters are discussed separately. Inthis work we make an attempt to: i) study the correlation between AUC and Cmax;ii) test the hypothesis of joint normality of log-transformed observations; iii) proposestatistical test for simultaneous equivalence. An real-data example was elaborated.

Keywords: Equivalence, Bioequivalence, Hypothesis testing, Simultaneous equiva-lence

Application of Mixture Regression Model inModeling Length of Stay in Hospital

A. Kazemnejad and M.R. Akhoond

Tarbiat Modares University

It is well known that empirical distribution of the Length of Stay in hospital (LOS)is positively skewed; researchers have attempted logarithmic transformation or otherforms of transformations to attain normality so that multiple regression and associa-tion test can be applied. Other researchers have recognized the heterogeneous natureof the LOS distribution, and proposed various trimming methods to distinguish out-liers of LOS. Such methods do not appear to logically justify. Their outlier thresholdsare either determined by distribution free measure or set arbitrary. Further, thesemethods often incur excessive trimming, which is inconsistent with the accepted no-tion of outliers as very unusual cases. Fetter (1984) proposed that patients with longhospital stay may have different component or distribution from other patients. Asa consequence, in this study instead of assuming a normal distribution for length ofhospital stay or adopting an arbitrary trim point for inpatient’s with long hospitalstay, we assume length of hospital stay as a mixture of weibull distribution, try tomodel it using weibull mixture model and use mixture of weibull regression modelsto study the impact of risk factors on the length of hospitalization.

Keywords:: Mixture Models, EM Algorithm, Inpatient’s Length of Hospital Stay

Robustness in Sequential Testing of Hypotheses onParameters of Random Sequences

Alexey Kharin

Belarusian State University

The sequential probability ratio test (SPRT) (A. Wald. Sequential Analysis, Wi-ley, New York (1947)) is used in many applications because of its optimal properties(A.N. Shiryaev. Statistical Sequential Analysis, Nauka, Moscow (1976)). The ob-served data are often contaminated, and this generates the necessity of robustness

20

analysis and robust sequential tests construction. We consider here two hypotheticalmodels under distortions. The cases of simple and composite hypotheses are analysed.1) Let i.i.d. random variables x1, x2, . . . be observed, xt ∈ U = 1, . . . ,M, from a dis-crete probability distribution with a parameter θ ∈ Θ = θ0, θ1: P (u; θ) = Pθxt =u = aJ(u;θ), u ∈ U , where a(1 is a fixed rational number, J(u; θ): U ×Θ → Z+ is afunction satisfying the condition:

∑u∈U aJ(u;θ) = 1. Consider the case of two simple

hypotheses: H0: θ = θ0, H1: θ = θ1. The SPRT is used for testing. Suppose thehypothetical model be contaminated: x1, x2, . . . be obtained from a mixture of dis-crete probability distributions P (u; θ) = (1− ε)P (u; θ) + εP (u; θ), u ∈ U ; ε ∈ (0, 1/2)is unknown, P (u; θ) is an unknown contaminating distribution, P (·) 6= P (·). In (A.Kharin. On Robustifying of the SPRT for a Discrete Model under “Contamina-tions””. Austrian Journal of Statistics, 31, p. 267-277 (2002)) we evaluate robustnessof the conditional error probabilities and expected sequence lengths for the SPRTand construct the robust sequential test for these distortions. Here we extend theresults to the family of arbitrary discrete probability distributions. We also considerthe case of absolutely continuous probability distributions of observations, proposean approximation for sequential tests, analyze the accuracy of such an approximationand construct the robust sequential test. The results on robustness analysis are alsoobtained in the paper for the case of composite hypotheses in the Bayesian setting.2) Let a homogeneous Markov chain x1, x2, . . . be observed, xt ∈ U , with an initialprobabilities vector π and a transition probabilities matrix P . Let there be two sim-ple hypotheses w.r.t. the parameters: H0: π = π(0), P = P (0), and H1: π = π(1),P = P (1); P (0) 6= P (1). We evaluate robustness of the conditional error probabilitiesand expected sequence lengths for sequential tests under the situation with distor-tions, where instead of hypothetical characteristics, the observed Markov chain hasthe initial probabilities vector and the transition probabilities matrix given by mix-tures π(k) = (1 − ε)π(k) + επ(k), P (k) = (1 − ε)P (k) + εP (k), k = 0, 1; π(k) 6= π(k)

and P (k) 6= P (k) are the vector of initial probabilities and the transition probabilitiesmatrix for the contaminating Markov chain. The robust sequential test is constructedw.r.t. the minimax risk criterion. The results are generalized for high-order Markovchains. We consider also the problem of sequential discrimination of autoregressivesequences, construct and analyze the robust sequential test. Numerical results illus-trate the theory. The research was supported by the Belarusian National ScienceFoundation, Project F06M-072.

Keywords: Robustness, Sequential analysis, Markov chain, Autoregression

Discrete Time Series with “Long Memory”

Yuriy Kharin


Mathematical modeling of complex systems and processes in genetics, economics,sociology and engineering needs adequate probability models for discrete time seriesxt ∈ A (t = 1, 2, . . . ; A = 0, 1, . . . , N − 1 is the finite set with 2 ≤ N(∞ elements)with “long memory”. A well known model for these discrete time series is the Markov

21

chain of high order s)1 defining the “memory depth”. Unfortunately, the number ofparameters Ds = Ns (N − 1) for this model increases exponentially w.r.t. the orders, and a researcher needs to observe a realization x1, x2, . . . , xn of inadmissibly largesize n)Ds to get good quality of statistical inferences. This situation generates verytopical problems of construction and statistical analysis of “low-parametric” high-order Markov chains (M. Waterman. Mathematical methods for analysis of DNAsequences. CRC Press, 1999), (A. Raftery. A model for high-order Markov chain.Journal of Royal Stat. Soc., 1985, Vol. 47, No.3, pp. 528-539), (Yu. Kharin. Markovchains with r-partial connections and their statistical estimation. Trans . of NationalAcademy of Sciences of Belarus, 2004, Vol. 48, No.1, pp. 40-44). The lecture isdevoted to the indicated topical problems and presents the following results:

• a review of the known “low-parametric” models of “long-memory” discrete timeseries;

• statistical estimators and tests for the discrete autoregressive time series DAR(s),and also results of the performance evaluation;

• results of statistical analysis based on the MTD-model (that was proposed byA. Raftery (A. Raftery. A model for high-order Markov chain. Journal of RoyalStat. Soc., 1985, Vol. 47, No.3, pp. 528-539) and on its generalizations;

• a new model — the high-order Markov chain of the order s with r partialconnections (1 ≤ r ≤ s) proposed in (Yu. Kharin. Markov chains with r-partialconnections and their statistical estimation. Trans . of National Academy ofSciences of Belarus, 2004, Vol. 48, No.1, pp. 40-44), and the results of statisticalanalysis for this model;

• results of numerical experiments on simulated and real data.

Keywords: Time series, ”Long memory”, Statistical analysis

A New Type of Stochastic Dependence Revealed inGene Expression Data

Lev Klebanov and Andrei Yakovlev

University of Rochester, USA

Modern methods of microarray gene expression data analysis are biased towardsselecting those genes that display the most pronounced differential expression. Themagnitude of differential expression does not necessarily indicate biological signif-icance and other criteria are needed to supplement the information on differentialexpression. Three large sets of microarray data on childhood leukemia were analyzedby an original method introduced in this paper. A new type of dependence betweenexpression levels in gene pairs was deciphered by our analysis. This modulation-likeunidirectional dependence between expression signals arises when the expression of a“gene-modulator” is stochastically proportional to that of a “gene-driver”. A total of

22

more than 35% of all pairs formed from 12550 genes were conservatively estimated tobelong to this type. Such pairs can form chains involving hundreds and even thou-sands of genes. In a given chain, each driver is at the same time a modulator of anotherupstream driver. However, this picture is not static: the composition of a particularchain of gene pairs may undergo dramatic changes when comparing two phenotypes.The ability to identify genes that act as “modulators” provides a potential strategy ofprioritizing candidate genes. This finding has the most direct implications for meth-ods of microarray data analysis that resort to pooling expression measures acrossgenes. In particular, we have studied a potential impact of the correlation betweengene expression levels and associated test statistics on the statistical inference basedon the nonparametric empirical Bayes methodology. We report evidence that thisimpact may be quite strong, leading to a high variance of the number of selectedgenes. Another disquieting effect is a high variance of the estimated false discoveryrate. The strength of dependence between genes has been gravely underestimated inthis field of biostatistical research.

Occupation Measure of Classical Risk Processes

Ekaterina T. Kolkovska

Centro de Investigacion en Matematicas Guanajuato, Mexico

We consider renewal risk processes with a continuous claim distribution G havingfinite mean. The questions we want to investigate are: What is the amount of time,before a given time horizon t¿0, during which the surplus stays in some fixed interval[a,b]? And if we define an expected recovering cost for the surplus process as inPicard (1994), how can it be calculated? Our answer to these questions is based ontwo random functionals of the risk process, namely its local time and its occupationtime. We prove existence of local time for the general class of renewal risk processeswith continuous claim distribution, and show that the distribution of the local timeat x up to time t is determined by the number of up-crossings at x performed by theprocess up to time t. For classical risk processes, we obtain the Laplace transform ofits occupation time in several important cases. As an application of our results, weobtain an explicit formula for the expected recovering cost of the classical risk processwith exponentially distributed claims, and give numerical illustrations.

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A semi-Markovian Branching Process on theClinical Cases for Modelling the Evolution of a

SEIR Disease in a Large Population. Example ofthe BSE Epidemics

Laurence Maillard-Teyssier Christine Jacob and Jean-Baptiste Denis

INRA, Jouy en Josas, FRANCE

We first build a semi-Markovian branching SEIR process in discrete time for mod-elling the evolution of BSE (Bovine Spongiform Encephalopathy) in a populationstructured by ages. Then, assuming a rare disease, we derive from this model a recur-sive stochastic process on the incidence of cases as the initial size of the populationincreases to the infinity. We show that this limit process belongs to the class of multi-type Bienayme’-Galton-Watson processes and that at each time, the number of casesis distributed according to a Poisson law conditionnally to the past. We give the bi-furcation parameter and calculate the extinction probability. We study the bayesianestimation and give credibility intervals for the contamination and incubation param-eters. These results are first applied to the observations of BSE cases in Great Britainbetween 1985 and 2005.

Keywords: semi-Markovian process, BSE, epidemiology, SEIR

Estimation and Testing a Nonlinear Transformationof the Integrated Processes

Vladimir Malugin


The paper is devoted to the problem of nonlinear transformation of linearly inte-grated process, which is closely connected with the nonlinear cointegration problem(Jin-Lung Lin, Clive W. J. Granger. Testing Nonlinear Cointegration. InternationalConference on Threshold Models and New Developments in Time Series (2004)). Toestimate the form of unknown nonlinear transformation tying together two time se-ries we consider two cases. In the first one we consider the class of differentiabletransformation functions for integrated of order one processes. For the given classof transformation we use a special stochastic approximation based on Taylor formulaand then test stationarity property for the reminder processes. If class of functionaltransformation is unknown we propose to use a nonparametric estimation procedurebased on a local kernel density estimates (V.I. Malugin. On the estimation of thedensity of random vectors with essentially dependent components. Vestnik of Belaru-sian State University, 2, p. 41?44 (1982)). The series of residuals is tested on thestationarity by means of KPSS test and other unit root tests to examine a nonlinearcointegration propety between given time series.

Keywords: Integrated and cointegated process, Nonlinear transformation, Nonpara-metric density estimation

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Uniformity obtained by Quasigroups: Part II

Smile Markovski and Verica Bakeva

University “Ss Cyril and Methodius”, Faculty of Natural Sciences and Mathematics,Institute of Informatics, P.O. Box 162, Skopje, Republic of Macedonia


Given a finite alphabet A and a quasigroup operation ∗ on the set A, in earlierpaper of ours we have defined the quasigroup transformation E : A+ → A+, whereA+ is the set of all finite strings with letters from A. Here we present several gen-eralizations of the transformation E and we consider the conditions under which thetransformed strings have uniform distributions of n-tuples of letters of A. The ob-tained results are very useful for application in cryptography, coding theory, definingand improving pseudo random generators, and so on.

Keywords: quasigroup, quasigroup string processing, uniform distribution

Multistate Systems with Graduate Failures andEqual Transition Intensities

Marija Mihova and Zaneta Popeska

”Ss Cyril and Methodius” University, Faculty of Natural Sciences and Mathematics,Institute of Informatics, P.O. Box 162, Skopje, Republic of Macedonia

e-mail: [email protected], [email protected]

We consider systems with n components in which the i-th component can work atMi + 1 linearly ordered levels of performance. The zero level is the level of completefailure and Mi is the level of perfect work of the component. We assume unrecoverablehomogenous systems with graduate failures, i.e. the underlying process of failure foreach component is a homogenous Markov process where the quality of the work of onecomponent can change only for one level lower than the observed one, and the failuresare independent for different components. Examining the properties of a subclass ofsuch systems where the transition intensities of all components and all levels are equal,we give a test for testing the hypothesis of equality. Under the assumption that theseintensities are equal we obtain a statistics for estimating the intensity of transition.

Keywords: Multi-state system, minimal path set, minimal cut set, failure intensities.

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Entropy Based Approach to Finding InteractingGenes Responsible for Complex Human Diseases

Valentin Milanov and Radoslav Nickolov

Fayetteville State University

A challenging problem in human genetics is the identification and characterizationof susceptibility genes for complex human diseases such as cardiovascular disease, can-cer, hypertension and obesity. These conditions are likely due to the effects of high-order interactions among multiple genes and environmental factors. Genome-wideassociation studies, where hundreds of thousands of single-nucleotide polymorphisms(SNPs) are genotyped in samples of cases and controls, offer a powerful approachfor mapping of complex disease genes. The classical statistical methods, parametricand non-parametric, are usually limited to small number of SNPs. Here we pro-pose a new method based on a classical search algorithm-sequential forward floatingsearch, utilizing entropy based criterion function. Using simulated case-control datawe demonstrate that the method has a high discovery rate under different models ofgene-gene interaction, including pure interaction without main effects of the genes.The performance of the proposed method is also compared to two methods recentlyadvocated in the literature: multifactor dimensionality reduction (MDR).

Keywords: complex disease, SNP, association.

A Simple Kernel-based Parameter Estimation

Kalin Milenkov1 and Nina Daskalova2

1FMI, Sofia University2IMI, Bulgarian Academy of Science

A part of an image recognition system, designed for limited computational re-sources is presented, describing a simple, ad hoc procedure for pose parameter esti-mation as a part of pose clustering algorithm with one- and two-dimensional accu-mulators.

Keywords: parameter estimation, pose clusteringThe paper is partially supported by the National Science Fund of Bulgaria, Grant NoVU-MI -105/2005.

26

Finite Time Blowup and Stability of a Reaction -Diffusion Equation with a Time-dependent Levy

Generator

Jose Alfredo Lopez Mimbela

Centro de Investigacion en Matematicas, Guanajuato, Mexico


Consider a reaction-diffusion equation of the form

∂u

∂t= Au + γuβ , u(0, x) = ϕ(x), x ∈ E, (1)

where A is the infinitesimal generator of a strong Markov process, E is a locallycompact space, γ)0 and β)1 are constants, and ϕ ≥ 0 is bounded and measurable. Itis well known that for any non-trivial initial value ϕ there exists a number such that(1) has a unique solution u on (0, Tϕ) × E, which is bounded on (0, T ) × E for any0(T (Tϕ, and if Tϕ(∞, then ‖u(t, ·)‖∞ →∞ as t ↑ Tϕ . When Tϕ = ∞ we say that uis a global solution, and when Tϕ(∞ we say that u blows up in finite time or that u isnonglobal. In the case of integer exponents β ≥ 2 it is known that the mild solutionto (1) can be represented in terms of an expectation functional of a related branchingparticle system, namely,

u(t, x) = E

eSt

∏y∈Bx(t)

ϕ(y)

, x ∈ E, t ≥ 0. (2)

Here (Bx(t))t≥0 is a branching particle system in E (with exponential individuallifetimes, branching numbers β, and particle motions with generator A), startingfrom an ancestor at x ∈ E, and St denotes the time length of the ancestor’s offspringtree up to time t ≥ 0. In this talk we use the above representation (2) to investigatethe semilinear non-autonomous Cauchy problem

∂u (t, x)∂t

= L(t)u (t, x) + γuβ (t, x) , t)s ≥ 0, u (s, x) = ϕ (x) , x ∈ Rd, (3)

where γ, β and ϕ are as in (1), and L(t), t ≥ 0, are Le’vy generators such that thecorresponding (time-inhomogeneous) martingale problem is well posed. We apply ourresults to generators of the form L(t) = k (t) ∆α, t ≥ 0, where k : (0,∞) → [0,∞) iscontinuous and does not vanish identically, and the operator ∆α is the generator of thed -dimensional symmetric α-stable process, 0(α ≤ 2. Assuming that

∫ t

0k(s) ds ∼ tρ

as t → ∞ for some ρ)0, we prove that the critical dimension for blow up of (3) isdc = α/(ρ(β − 1)), meaning that if d(dc, then, apart from u ≡ 0, there are no positiveglobal solutions to (3), whereas if d)dc, then the solution u of (3) is global providedu(0, ·) is sufficiently small. We also show that the case ρ = 0, which corresponds toan integrable k, yields finite time blow up of u for any nontrivial initial value.

Keywords: Galton-Watson trees, Semilinear partial differential equations, Feynman-Kac representation

27

Reinsurance by the Polya - Aeppli Risk Model

Leda D. Minkova

Faculty of Mathematics and Informatics Sofia University ”St. Kl. Ohridski”


We consider the Polya - Aeppli risk model, defined in (1). Suppose the insurerhas the possibility to choose proportional reinsurance and the premium loading de-pends on the retention level. In this note we show that the optimal retention levelcan be found. In the small claim case there is a unique retention level maximizingthe adjustment coefficient and one implicit equation has to be solved. The effect ofreinsurance on the probability of ruin is discussed.(1) Minkova L.D. (2004) The Polya-Aeppli process and ruin problems, JAMSA,2004(3), 221 - 234.

Keywords: Polya - Aeppli risk model, ruin probability, reinsurance.

An Option Pricing Formula based on BranchingProcesses

Georgi Mitov1 and Kosto Mitov2

1Dept of Probability and Statistics, IMI, BAS, Sofia, Bulgaria2Aviation Faculty - NMU 5856 D. Mitropolia, Pleven, Bulgaria

On a probability space (Ω, A, P ), assume the following: (i) The Galton-Watsonbranching process Zn, n = 0, 1, 2, ...; (ii) The integer valued subordinator N(t), t ≥ 0.Epps introduced the process P (t) = ZN(t), t ≥ 0 as a model for stock prices. Using thisresult, we derive a formula for pricing of European call options. Some new propertiesof the process in the supercritical case are proved too. References 1. Epps, T., (1996),Stock prices as branching processes, Communications in statistics: Stochastic Models,12, 529-558.

Keywords: branching processes; random time; option pricing

Sum and Extremal Processes over Explosion Area

Ivan Mitov1 and Elisaveta Pancheva2

1FMI, Sofia University2IMI - BAS

Given a simple in time Poisson point process (P.p.p.) N = (Tk, Xk) : k ≥ 1on Z = (0,∞) × (0,∞)d with mean measure (m.m.) µ we construct the associatedextremal process Y by Y (t) = C(t) ∨ ∨Xk : Tk ≤ t Here C : (0,∞) → (0,∞)d

is the lower curve of Y . It is increasing and right continuous. Denote by f(t, x)

28

the distribution function of the process Y (t). There is a direct connection betweenf and the m.m. µ of N , namely f(t, x) = exp−µ((0, t) × (0, x)c) Denote by Athe set (0, C)c. We observe that µ((0, t) × (0, x)c) is finite for all (t, x) in A and= ∞ for all (t, x) in (0, C). The set (0, C) is ”explosion area” of µ. We delete thepoints (Tk, Xk) which lie below the curve C. Then we have p.p. N with points inA. With N we associate also a sum process S on A by S(t) = C(t) ∨

∑Xk : Tk ≤ t

Let Nn = (Tnk, Xnk) : k ≥ 1 be Bernoulli p.p. on Z for all n ≥ 1 with countingprocess Nn(t) = maxk : Tnk ≤ t and let Yn and Sn be the associated extremal andsum process. In this contribution we are interested in the weak convergences on A:Yn → Y , Sn → S and Nn → N , and their relationships. The approach, used here isthrough the ”accompanying” p.p.’s N

(a)n = (tnk, Xnk) : k ≥ 1 , n ≥ 1 with counting

functions kn(t) = maxk : tnk ≤ t and associated extremal Y(a)n and sum S

(a)n

processes. Both p.p.’s Nn and N(a)n are connected by the relation Nn(t) = kn(θn(t))

where θn : (0,∞) → (0,∞) is random time change.

Keywords: Extremal process, Sum Process, Point process, Weak convergence

Controlled Multitype Branching Models: GeometricGrowth

Manuel Mota Miguel Gonzalez Rodrigo Martinez

Department of Mathematics, University of Extremadura

The multitype Galton-Watson process is a well-known branching model which hasreceived considerable attention in the scientific literature (e.g. see Mode (1971)).From this model, it is possible to obtain other homogeneous branching models for amore suitable description of some real situations. One can think of different mod-ifications by adding new features to the original multitype Galton-Watson process:- To consider population-size-dependent reproduction. - To establish a control ofthe number of each type of progenitor according to the population size. - To al-low interaction between individuals of the same generation at reproduction time, i.e.”dependent offspring”. Control in the population was proposed by Sevast’yanov andZubkov (1974) in a deterministic way and population-size-dependent reproduction wasconsidered by Klebaner (1989). Gonzalez et al. (2005) have introduced a multitypemodel that puts together control and size dependent reproduction, generalization ofthe one-dimensional model with control and reproduction dependence on the popu-lation size considered by Kuster (1985). In this model, called Controlled MultitypeBranching Process with Random Control and Population-Size-Dependent Reproduc-tion (CMPD), the number of progenitors of each type is controlled by means of arandom mechanism and possible dependence among individuals of the same genera-tion at reproduction time is allowed. The introduction of dependence is a major novelfeature with respect to the classical branching models, since their implicit assumptionof independence can be only considered to be a mere theoretical simplification of themore complex types of reproductive behaviour in nature. Under certain assumptions(see Gonzalez et al. (2005)) a CMPD has a positive probability of indefinite growth.

29

In this work we provide both sufficient and necessary conditions for this growth to begeometric. Specifically we present results about the almost sure and Lα, 1 ≤ α ≤ 2,convergence of the CMPD, normed by a geometric progression, to a non-degenerateand finite random vector.References:Gonzalez, M., Martinez, R., Mota, M., (2005), On the unlimited growth of a class ofhomogeneous multitype Markov chains. Bernoulli, 11 (3), 559-570.Klebaner, F. (1989), Linear growth in near-critical population-size-dependent multi-type Galton-Watson processes. J. Appl. Probab., 26, 431-445.Kuster, P. (1985), Asymptotic growth of controlled Galton-Watson processes. Ann.Prob., 13, 1157-1178.Mode, C. (1971), Multitype branching processes. New York: Elsevier.Sevast’yanov, B.A., Zubkov, A. (1974), Controlled branching processes. Theor. Prob.Appl., 19, 14-24.The research was supported by the Ministerio de Ciencia y Tecnologia and the FEDERthrough the Plan Nacional de Investigacion Cientifica, Desarrollo e Innovacion Tec-nologica, grant BFM 2003-06074.

Stability of the Inventory - Backorder Process inthe (R,S) Inventory/Production Model

Zahir Mouhoubi and Djamil Aissani

Laboratory of Modelization and Optimization of Systems

The aim of this talk is to obtain the sufficient conditions and some estimates ofthe uniform ergodicity and the strong stability of the inventory-backorder process in asingle-item, single location, (R,S) inventory/production model with limited capacityof production per period and uncertain demands. In this order some intermediateresults are established and an overview about the main stability methods for stochasticprocesses and the performance measure in the inventory models are also considered.

Keywords: Strong stability, (R,S) policy, inventory/production model, Uniformergodicity

Extraction of Fraud Schemes from Trade Series

Charalambos Moussas and Veska Noncheva

TP-361, DG JRC Ispra Site, European Commission, Via E. Fermi 1, Ispra(VA), I-21020,

ITALY

It is very often the case that the patterns of a fraudulent activity in trade arehidden within existing trade data time series. Furthermore, with the advent of pow-erful and affordable computing hardware, relatively big amounts of available tradedata can be quickly analyzed with a view to assisting anti-fraud investigations in this

30

field. In this paper, based on the availability of such import/export data series, wepresent a statistical method for the identification of potential fraud schemes, by ex-tracting and highlighting those cases which lend themselves to further investigationby anti-fraud domain experts. The proposed method consists in applying time seriesanalysis for prediction purposes, calculating the resulting significant deviations, andfinally clustering time series with similar patterns together, thus identifying suspector abnormal cases.

Keywords: Fraud Detection, Change Detection, Time Series Analysis, Forecasting,Cluster Analysis

The Number of Parts of Given Multiplicity in aRandom Integer Partition

Ljuben Mutafchiev1 and Emil Kamenov2

1American University in Bulgaria and Bulgarian Academy of Sciences2Sofia University

Random partition of integers are treated in the case where all partitions of anpositive integer n are assumed to have the same probability. Let X(m,n) denote thenumber of parts of multiplicity m in a random integer partition. We study the asymp-totic behaviour of the mean and the standard deviation of X(m,n) as n tends to infin-ity. Using moment generating functions, we also establish that X(m,n), appropriatelynormalized, tends weakly to the standard normal distribution.

Keywords: Integer partitions

Robust Fitting of Mixture Models

N. Neykov1 P. Filzmoser R. Dimova and P. Neytchev1

1National Institute of Meteorology and Hydrology, Bulgarian Academy of Sciences

An overview of the robust techniques for fitting mixture of distributions to datawill be made. Special attention will be devoted to the performance of the WeightedTrimmed Likelihood Estimator (WTLE) defined in Vandev and Neykov (1998). Thebreakdown point properties of the WTLE in mixture settings will be discussed fol-lowing Mueller and Neykov (2003). The FAST-TLE algorithm of Neykov and Mueller(2003) will be adapted to get approximate parameter estimates of the mixture com-ponents. Examples of real and artificial data will be used to illustrate the superiorityof the TLE versus the MLE approach.References:D.L. Vandev and N.M. Neykov (1998). About regression estimators with high break-down point, Statistics, 32, 111-129.C.H., Mueller and N.M. Neykov (2003). Breakdown points of the trimmed likelihood

31

and related estimators in generalized linear models. J. Statist. Plann. Inference 116,503-519.N.M. Neykov and C.H. Muller (2003). Breakdown point and computation of trimmedlikelihood estimators in generalized linear models. In: Dutter, R., Filzmoser, P.,Gather, U., Rousseeuw, P.J. (Eds.), Developments in robust statistics. Physica-Verlag, Heidelberg, 277-286.

Keywords: Mixture of distributions, Trimmed Likelihood Estimator, Robustness,Breakdown Point

Modelling Daily Precipitation over the Territory ofBulgaria using Hidden Markov Models

N. Neykov W. Zucchini P. Neytchev and H. Hristov

National Inst. of Meteorology and Hydrology, Bulgarian Academy of Sciences

Homogeneous and non-homogeneous hidden Markov models (McDonald and Zuc-chini, 1997) for relating precipitation occurrence and amounts at a network of rainfallstations broadly covering the territory of Bulgaria to large-scale atmospheric circu-lation patterns will be presented. The quality of the developed models are assessedby comparison of model simulated to historical rainfall statistics. Data handling andmodel fitting are carried out in the R environment.References:McDonald, J. and Zucchini, W. (1997). Hidden Markov and Other Models forDiscrete-valued Time Series. Chapman and Hall, London.

Keywords: Daily precipitation, Hidden Markov model

A Test of Association Between Qualitative Traitand a Set of SNPs

Radoslav Nickolov and Valentin Milanov

Fayetteville State University

In this work we propose a novel candidate gene association test utilizing a set oftightly linked single nucleotide polymorphisms (SNPs). This is a powerful likelihoodratio test, based on Gibbs random field model. We use simulation studies to evaluatethe type I error rate of our proposed test, and compare its power with that of othercandidate gene association tests.

Keywords: SNP, association test, Gibbs distribution, likelihood ratio

32

Scatter Matrices, Kurtosis and IndependentComponents

Hannu Oja Seija Sirkia and Jan Eriksson

Tampere School of Public Health, University of Tampere

In the independent component analysis (ICA) it is assumed that the componentsof the multivariate independent and identically distributed observations are lineartransformations of latent independent components. The problem then is to find the(linear) transformation which transforms the observations back to independent com-ponents. In the talk the ICA is discussed and it is shown that, under some mildassumptions, two scatter matrices may be used together to find the independent com-ponents. The independent components are then given in the kurtosis order. Thescatter matrices must then have the so called independence property. The theory isillustrated by examples.

Keywords: Affine Equivariance, Elliptical Model, Independent Component Analysis,Principal Component Analysis

A Comparative Performance Analysis of theTurkish Industrial Sectors in Terms of ERP Success

Ali Ihsan Ozdemir

Erciyes Universtiy, Economics & Administrative Science Faculty Management Department

Kayseri, Turkey

In this study industrial sectors in Turkey are examined in terms of ERP (EnterpriseResource Planning) Success. Sectors are represented by firms that have been appliedand used ERP systems. To evaluate the sectors we used 7 performance metrics groupcontaining 28 metrics that can be used in determination of ERP systems success.These metrics are related with product, production process, costs, other financialindicators, delivery, supply and customers. Industrial sectors represented by firmsevaluated by multivariate statistical analiysis techniques such as ANOVA and FactorAnalysis for their ERP success. Firms are selected from top 500 firms of Turkey interms of their production to sells.

Keywords: Turkish Industrial Sectors, ERP, Performance Evaluation, MultivariateStatistical Analysis, Factor A

33

Generalized Split-BREAK Process

Biljana Popovic1 and Vladica Stojanovic2

1Faculty of Sciences and Mathematics, University of Nis2Faculty of Economics, University of Pristina in Kosovska Mitrovica

We present the modification of so called STOPBREAK process. The initial worksin this region were published by Engle and Smith (1999) and Gonzales (2004). Thecontribution of Popovic and Stojanovic (2005) is the model of ARCH type, Split-ARCH. Now we present the process that is the generalization of our threshold STOP-BREAK process. We define it as

A(L)yt = B(L)qtεt + C(L)(1− qt)εt , t ∈ Z

where L is the shift operator, qt is the noise indicator, A(L) = 1−∑m

i=1 αiLi, B(L) =

1−∑n

j=1 βjLj and C(L) = 1−

∑pk=1 γkLk.

We investigate the stochastic properties of this process and apply it on some log-volumes of the real data presented on Belgrade Stock Exchange.

Keywords: Split-BREAK, log-volumes

Interactive Regularized Discriminant Analysis forDiscriminating Authentic Wines from five Countries

U. Romisch1, H. Jager1 and D. Vandev2

1TU Berlin, Fak. III, Gustav- Meyer- Allee 25, D- 13355 Berlin2Univ. of Sofia, Blvd. Bourchier 5, B- 1164 Sofia

Project Steering Committee: R. Wittkowksi BfR, Germany, P. Brereton CSL, United

Kingdom, E. Jamin Eurofins, France, X. Capron VUB, Belgium, C. Guillou JRC, Italy, M.

Forina UGOA, Italy, U. Romisch TUB, Germany, V. Cotea UIASI.VPWT.LO, Romania, E.

Kocsi NIWQ, Hungary, R. Schoula CTL, Czech Republic.

The determination of the geographical origin (country) of wines on the base ofchemico-analytical parameters was the aim of the European project ”Establishing ofa wine data bank for analytical parameters for wines from Third countries (G6RD-CT-2001-00646-WINE DB)”, supported by the European Commission. Therefore awine data base containing 1200 samples of commercial and authentic wines from Hun-gary, Czech Republic, Romania and South Africa over a period of three years has beencreated. To the second-year-data could be added 50 authentic wines from Australia.For each of those samples around 100 analytical parameters, among them rare earthelements and isotopic ratios were measured. After analysing first-year-data thesenumber could be reduced to 64 parameters. Besides other multivariate statisticalmethods of discrimination and classification the method of Regularized DiscriminantAnalysis (RDA) was used to distinguish the wines of the different countries on thebase of a minimal number of the most important parameters. The method of RDA

34

as well as the used MATLAB-program, developed by Vandev, which allows an in-teractive stepwise discriminant model building on the base of an optimal choice ofthe ”nonlinearity” parameter alpha will be described shortly. Taking the authenticsecond-year-data, including Australian wines as a basis, discriminant models with cor-responding classification and prediction error rates will be given. As a result of usingRDA, the number of analytical parameters could be reduced to the most importantnecessary parameters (5-7) for recognizing the geographical origin of authentic wines.

Some Probabilistic Results in a Bisexual BranchingProcess with Immigration

Alfonso Ramos Manuel Molina and Ines del Puerto

Department of Mathematics, University of Extremadura, 10.071, Caceres, Spain.

With the main objective to describe the probabilistic evolution of two-sex popula-tions, where females and males coexist and form couples (female-male mating units),some bisexual branching processes have been investigated. We refer the reader to [1],or [2], for surveys about these processes. In this work, we introduce a discrete-timebisexual branching process which considers mating, offspring, and females and malesimmigration dependent on the number of couples in the population at the previousgeneration. We define the bisexual branching process with females and males immi-gration as a stochastic model (Fn,Mn)n≥1 defined, for n = 0, 1, . . ., in the recursive

form: (Fn+1,Mn+1) =Zn∑i=1

(fn,i(Zn),mn,i(Zn)) + (f In+1(Zn),mI

n+1(Zn)),

Z0 = N0 ≥ 1, Zn+1 = LZn(Fn+1,Mn+1),

where, for N ∈ Z+, (fn,i(N),mn,i(N))n≥0;i≥1 and (f In+1(N),mI

n+1(N))n≥1 areindependent sequences. Each one of these sequences is formed by independent, iden-tically distributed, non-negative, and integer-valued random variables. LNN≥0 isa sequence of non-negative real functions on R+ ×R+ such that each LN is assumedto be monotonic non-decreasing in each argument, integer-valued on the integers,and satisfying that LN (x, 0) = LN (0, y) = 0, x, y ∈ R+, with Z+ and R+ denoting,respectively, the non-negative integer and real numbers. For this stochastic model,we establish some transition results, relationships among its probability generatingfunctions and stochastic monotony properties.References:1. P. Haccou, P. Jagers and V. Vatutin. Branching processes: Variation, growth, andextinction of populations. Cambridge University Press, 2005.2. D. M. Hull. A survey of the literature associated with the bisexual Galton-Watsonbranching process. Extracta Math., 18: 321–343, 2003.Keywords: branching processes, bisexual processes, immigration processes, population-size dependent processes.

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Size-biased Branching Processes with OverlappingGenerations

Serik Sagitov

Chalmers University of Technology

We discuss a size-biased version of the Crump-Mode-Jagers branching processbased on the stable pedigree law described by Jagers and Nerman.

On the Use of Higher-Order Statistical Tests inSignal Processing

Blagovest Shishkov

Institute of Mathematics & Informatics, Bulgarian Academy of Sciences

Acad. G. Bonchev Str., Bl. 8 Sofia 1113, Bulgaria

Tests of hypotheses based on Higher-Order Statistics (HOS) are reviewed in theparticular context of the identification of nonlinear processes. A general and uni-fied procedure is suggested in order to construct statistical tests: (1) for detectinga non-gaussian signal in a gaussian (or non-gaussian) noise, (2) testing a stochastictime series for non-gaussianity (including non-linearity), (3) studying non-linear phe-nomena by using the kth-order coherency function. Asymptotic theory of estimatesof the kth- order spectra is implemented in a digital signal processing framework.

Keywords: kth-order test-statistic, spectrum, non-linearity, non-gaussianity, signaldetection and estimation

Limit Theorems for Maxima of Heavy-tailed Termswith Random Dependent Weights

Stilian A. Stoev and Murad S. Taqqu

Let Uj , j = 1, 2, ... be independent identically distributed random variables withheavy, regularly varying tails. The theory about the limit behavior of the maximamaxUj , j = 1, .., n, as n tends to infinity is well developed. Here, we consider asequence of non-negative weights Wj , j = 1, 2, ... and focus on the weighted maximaMn(t) := maxWjUj , j = 1, .., (nt) where Mn(t) = W1U1 if t is in (0, 1/n). Herethe sequences Uj and Wj are assumed to be independent. We study the generalcase when the weights Wj , j = 1, 2, ... can be dependent and in particular long-rangedependent. Under mild tail and convergence conditions on the weights Wj ’s, weestablish limit theorems for scaled versions of the process Mn(t), as n tends toinfinity. The limit processes are mixtures of extremal Frechet processes. The resultsare valid when the laws of the Uj’s belong to the normal domain of attraction of aFrechet distribution or to a large sub-class of the general domain of attraction of aFrechet law.

36

Keywords: weighted maxima, random weights, limit theorems, extremal Frechetprocess

Smoothed Density Estimation of Interval CensoredData

Eugenia Stoimenova

Institute of Mathematics & Informatics, Bulgarian Academy of Sciences

We consider the problem of estimating a probability density function based ondata that are incomplete due to interval censoring. The (nonparametric) maximumlikelihood estimator for the corresponding distribution function is well defined. Forthe density function this is not the case. We study two nonparametric estimators forthis density. The first is a type of kernel density estimate based on the conditionalexpectation of a kernel over these intervals. The second is a modification of Nadaraya-Watson kernel density estimator which allow censored data.

MSC 2000: *62G07 Curve estimation, *62F30 Statistical inference under con-straints

Levy Processes and some Generalizations

Pavel Stoynov

Sofia University ”St. Kliment Ohridski”

The Levy processes, which include Poisson process and Brownian motion as spe-cial cases, were the first class of stochastic processes to be studied in the modernspirit (by the French mathematician Paul Levy). In the paper, the author considerssome generalizations of this class of processes. At a given filtered probability spacesatisfying the usual hypotheses an adapted process with independent and station-ary increments X(t) is called generalized Levy process if it has the representationX(t) = L(t) + J(t) where L(t) is a Levy process and J(t) is a sum of independentrandom jumps at times Ti. The set Ti is either a sequence of stopping times (thenthe defined class of processes is called generalized Levy process of class 2 and is de-noted GL2) or a deterministic discrete set (then the defined class of processes is calledgeneralized Levy process of class 3 and is denoted GL3). The processes with inde-pendent and stationary increments also can be considered as generalizations of Levyprocesses denoted GL1. Then GL3 is a subset of GL2 and GL2 is a subset of GL1.For a generalized Levy processes, Fourier transform, some representations and someother properties are considered.

Keywords: Levy Processes

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Histogram Density Estimator for Censored Data

Yonko Stoynov1 and Eugenia Stoimenova2

1Technical University, Sofia2Institute of Mathematics & Informatics, Bulgarian Academy of Sciences

In this paper we treat the problem of nonparametric estimation of density functionfor interval censored data. Definition and properties of histogram density are given.Kernel density and iterative algorithm are used to obtain new estimates.

Two Random Variables Dependence Function

Nataliya Toncheva

Shoumen University ”Bishop Konstantin Preslavski”

A new concept of dependence function is offered in the paper. It is possible tomeasure the intense and the nature of dependence of two random variables by usingthat function. The dependence function gives much more complete information of thedependence between two random variables than the usual correlation and regressionratios. Some problems related with the new concept of dependence function areoffered.

Keywords: dependence function, random variables, problems

Markov Chain Order Estimating and itsApplications in Bioinformatics

Ivan Tsitovich Evgeniya Naumova Fyodor Tsitovich

Institute for information transmission problems 127994, GSP-4, Russia, Moscow, BolshoiKaretnyi per., 19


We investigate an upper bound of Markov chain order for its consistent estimateexistence. It is found that in contrast with theoretical results for practical problemsthe Markov chain order can not be large. The upper bound of Markov chain orderfor statistical analysis of protein sequences is received.

Keywords: Markov chain order, consistent estimate, bioinformatics.

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On Robust Models of Multyservice System Traffic

Ivan Tsitovich Yurij Bubnov Eleonora Melik-Gaykazova

Institute for information transmission problems 127994, GSP-4, Russia, Moscow, BolshoiKaretnyi per., 19


A multiservice network with a mixture of independent self-similar diffusion pro-cesses as an input flow is investigated. It is found that a cell loss probability is notrobust characteristic as a function of potions of self-similar diffusion processes withdifferent Herst parameter in the input flow. A problem of an input traffic control forprescribed parameters of the network QoS is considered.

Keywords: robust model, multyservice network, fractal processes

An Educational Technology for the Formation ofTranscendental Thinking

Anna Veleva

University of Plovdiv, Bulgaria

The author offers an educational technology of kinematically tracing the processesof learning, understanding, assessment and possibilities for a successful mastering of aboundless, contradictory and hard to predict situation. This method is consistent withthe laws of realization and helps the process of language universalization. It makesthe primary concepts and axioms in Theory of Probabilities intuitively familiar.

Keywords: experience, elementary outcome of a certain experiment, equal possibilityof all elementary outcomes

Joint Densities of Correlation Coefficients forSamples from Multivariate Standard Normal

Distribution

Evelina I. Veleva

Rouse University, Rouse

We consider the joint distribution of the correlation coefficients for samples frommultivariate standard normal distribution. Some marginal densities are obtained.Independence and conditional independence between sets of sample correlation coef-ficients are established.

Keywords: Multivariate normal distribution, sample correlation coefficients, inde-pendence, conditional independence.

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Linear Regression with Poisson Error

Mohammad Luay Wakeel

Sofia University

The linear models are fitted by likelihood or least-squares method. These methodsare applied when the models have Gaussian errors or the depended variable and errorhave the same distribution from the exponential family. In this paper the linearmodel has Poisson error but the dependent variable has unknown distribution.TheEM-Algorithm is used here for fitting such models.

Keywords: EM-Algorithm, Poisson Error

Branching Processes and Cell Proliferation withContinuous Labeling

A. Y. Yakovlev1 and N. M. Yanev2

1Department of Biostatistics and Computational Biology, University of Rochester, 601Elmwood Avenue, Box 630, Rochester, New York 14642, U.S.A.

e-mail: Andrei [email protected] of Probability and Statistics, Institute of Mathematics and Informatics,

Bulgarian Academy of Sciences, 8, G. Bonchev, Sofia 1113, Bulgariae-mail: [email protected]

It is considered an age-dependent branching process, where the particles are char-acterized with a continuous parameter. This process can be interpreted as a modelof cell proliferation with continuous labeling. The equations for pgf and the momentare obtained. It is defined the so-called label distribution. The label distribution iscalculated explicitly for some non-markov particular cases, which are interesting forthe cell proliferation. Finally the Markov case is considered with more details. Notethat the discrete label analog is investigated by N.M.YANEV and A.Y.YAKOVLEV(1985) On the distribution of marks over a proliferating cell population obeying theBellman-Harris branching process. Math. Biosci. 75, no.2. 159-173.

Keywords: branching processes; cell proliferation; continuous labeling

On Extreme Value Results in Branching Processes

George P. Yanev

University of South Florida, USA

We focus our attention to some recent results concerning extremes in branchingprocesses. In particular, we review studies on maximum family size in processes withand without varying environments, and extremes in branching trees. The methods

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and techniques used in these studies are outlined. The results to be presented areclosely related to the following publications:1. Number of complete N-ary subtrees on Galton-Watson family trees, Methodologyand Computng in Applied Probability, in press (joint with L. Mutafchiev).2. Extremes of Geometric Variables with Applications to Branching Processes. Statis-tics and Probability Letters, 65(2003), 4:379-388 (joint with K. Mitov and A. Pakes).3. Family size order statistics in branching processes with immigration. StochasticAnalysis and Applications, 18 (2000), 4:655-670, (joint with C.P. Tsokos).4. On the maximum family size in branching processes. J. Applied Probability, 36(1999), 3:632-643 (joint with I. Rahimov).

Keywords: branching processes, extremes, varying environments, random trees

Branching Symmetric Random Walk ond-dimensional Lattice

E. Yarovaya

Department of Probability Theory, Moscow State University

We study a symmetric continuous time branching random walk on d-dimensionallattice with finite variance of jumps under the assumption that the birth and the deathof particles occurs at a single site (i.e., the source). In the critical and subcriticalcases asymptotics is established for the survival probability of particles on lattice.Conditional limit theorems for the population size are proved.

Keywords: critical case, subcritical case

Branching Processes and Insurance

Gheorghita Zbaganu

Bucharest University, Romania


Somebody (conventionally denoted by ”o”) wants to make a fund to insure all itsoffspring with 1 MU at the end of birth year. Suppose that we accept the classicali.i.d. assumptions and, moreover, that we know the interest rate of the fund. Howgreat should be the fund ?

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